Probabilistic Roadmap Methods (PRMs) Nancy M. Amato Parasol Lab,Texas A&M University

Motion Planning The Alpha Puzzle The Piano Movers Problem start goal obstacles (Basic) Motion Planning: Given a movable object and a description of the environment, find a sequence of valid configurations that moves it from the start to the goal. Have movies for both cases - you can play them as you speak.

Applications of PRM-based Motion Planning (Amato et al)

Outline C-space, Planning in C-space (basic definitions) Probabilistic Roadmap Methods (PRMs) PRM variants (OBPRM, MAPRM) User-Input: Haptically Enhanced PRMs PRMs for more `complex’ robots deformable objects group behaviors for multiple robots protein folding

Configuration Space (C-Space) “robot” maps to a point in higher dimensional space parameter for each degree of freedom (dof) of robot C-space = set of all robot placements C-obstacle = infeasible robot placements C-obst 3D C-space (a,b,g) a b g 3D C-space (x,y,z) 6D C-space (x,y,z,pitch,roll,yaw) 2n-D C-space (f1, y1, f2, y2, . . . , f n, y n)

C-obstacles workspace obstacles transformed to C-obstacles Figure from Latombe’91

C-obstacles => complicated 3D C-obstacle simple 2D workspace obstacle => complicated 3D C-obstacle Figure from Latombe’91

Motion Planning in C-space Simple workspace obstacle transformed Into complicated C-obstacle!! C-space Workspace C-obst C-obst Path is 1D curve Path is swept volume obst obst C-obst C-obst obst obst y x robot robot

The Complexity of Motion Planning Most motion planning problems of interest are PSPACE-hard [Reif 79, Hopcroft et al. 84 & 86] The best deterministic algorithm known has running time that is exponential in the dimension of the robot’s C-space [Canny 86] C-space has high dimension - 6D for rigid body in 3-space simple obstacles have complex C-obstacles impractical to compute explicit representation of freespace for more than 4 or 5 dof So … attention has turned to randomized algorithms which trade full completeness of the planner for probabilistic completeness and a major gain in efficiency

Roadmap Construction (Pre-processing) Probabilistic Roadmap Methods (PRMs) [Kavraki, Svestka, Latombe,Overmars 1996] C-space Roadmap Construction (Pre-processing) Query processing start goal 1. Randomly generate robot configurations (nodes) - discard nodes that are invalid 1. Connect start and goal to roadmap C-obst 2. Connect pairs of nodes to form roadmap - simple, deterministic local planner (e.g., straightline) - discard paths that are invalid 2. Find path in roadmap between start and goal - regenerate plans for edges in roadmap C-obst C-obst C-obst C-obst Primitives Required: Method for Sampling points in C-Space Method for `validating’ points in C-Space

PRMs: Pros & Cons PRMs: The Good News PRMs: The Bad News 1. PRMs are probabilistically complete 2. PRMs apply easily to high-dimensional C-space 3. PRMs support fast queries w/ enough preprocessing Many success stories where PRMs solve previously unsolved problems C-obst start goal PRMs: The Bad News 1. PRMs don’t work as well for some problems: unlikely to sample nodes in narrow passages hard to sample/connect nodes on constraint surfaces Our work concentrates on improving PRM performance for such problems. start goal C-obst

Related Work (selected) Probabilistic Roadmap Methods Uniform Sampling (original) [Kavraki, Latombe, Overmars, Svestka, 92, 94, 96] Obstacle-based PRM (OBPRM) [Amato et al, 98] PRM Roadmaps in Dilated Free space [Hsu et al, 98] Gaussian Sampling PRMs [Boor/Overmars/van der Steppen 99] PRM for Closed Chain Systems [Lavalle/Yakey/Kavraki 99, Han/Amato 00] PRM for Flexible/Deformable Objects [Kavraki et al 98, Bayazit/Lien/Amato 01] Visibility Roadmaps [Laumond et al 99] Using Medial Axis [Kavraki et al 99, Lien/Thomas/Wilmarth/Amato/Stiller 99, 03, Lin et al 00] Generating Contact Configurations [Xiao et al 99] Single Shot [Vallejo/Remmler/Amato 01] Bio-Applications: Protein Folding [Song/Thomas/Amato 01,02,03, Apaydin et al 01,02] Lazy Evaluation Methods: [Nielsen/Kavraki 00 Bohlin/Kavraki 00, Song/Miller/Amato 01, 03] Related Methods Ariadnes Clew Algorithm [Ahuactzin et al, 92] RRT (Rapidly Exploring Random Trees) [Lavalle/Kuffner 99]

Other Sampling Strategies Obstacle Sensitive Sampling Strategies Goal: Sample nodes in Narrow Passages OBPRM: Obstacle-Based PRM sampling around obstacles MAPRM: Medial-Axis PRM sampling on the medial axis Repairing/Improving approximate paths transforming invalid samples into valid samples

OBPRM: An Obstacle-Based PRM [IEEE ICRA’96, IEEE ICRA’98, WAFR’98] To Navigate Narrow Passages we must sample in them most PRM nodes are where planning is easy (not needed) start goal C-obst Idea: Can we sample nodes near C-obstacle surfaces? we cannot explicitly construct the C-obstacles... we do have models of the (workspace) obstacles... OBPRM Roadmap PRM Roadmap start goal C-obst

OBPRM: Finding Points on C-obstacles 2 4 Basic Idea (for workspace obstacle S) 5 3 1. Find a point in S’s C-obstacle (robot placement colliding with S) 2. Select a random direction in C-space 3. Find a free point in that direction 4. Find boundary point between them using binary search (collision checks) Note: we can use more sophisticated heuristics to try to cover C-obstacle 1 C-obst

PRM vs OBPRM Roadmaps PRM 328 nodes 4 major CCs OBPRM 161 nodes

OBPRM for Paper Folding Box: 12 => 5 dof Periscope: 11 dof Roadmap Statistics 218 nodes, 3 sec Roadmap Statistics 450 nodes, 6 sec

MAPRM: Medial-Axis PRM Steve Wilmarth, Jyh-Ming Lien, Shawna Thomas [IEEE ICRA’99, ACM SoCG’99, IEEE ICRA’03] Key Observation: We can efficiently retract almost any configuration, free or not, onto the medial axis of the free space without computing the medial axis explicitly. Medial axis C-obstacle

Repairing Approximate Paths Problem: Even with the best sampling methods, roadmaps may not contain valid solution paths may lack critical cfgs in narrow passages may contain approximate paths that are ‘nearly’ valid Repairing/Improving Approximate Paths repaired path Create initial roadmap Extract approximate path P Repair P (push to C-free) Focus search around P Use OBPRM-like techniques C-obstacle approximate path C-obstacle

Hybrid Human/Planner System [IEEE ICRA’01, IEEE ICRA’00, PUG’99] PHANToM 1. User collects approximate path using haptic device User insight identifies critical cfgs User feels when robot touches obstacles and adjusts trajectory 2. Approximate path passed to planner and it fixes it Current Applications Intelligent CAD Applications Molecule Docking in drug design Animation w/ Deformable Models

Hybrid Human/Planner System Haptic Hints Results 0.85 0.95 1 Flange Problem

Hybrid Human/Planner System Haptic Hints Results 0.85 0.95 1 Flange Problem

Applications Same Method: Diverse range of problems Deformable Objects Simulating Group Behaviors homing, covering, shepherding Modeling Molecular Motions Protein folding, RNA folding, protein/ligand binding

Deformable Objects [IEEE ICRA’02] Problem: Find a path for a deformable object that can deform to avoid collision with obstacles deformable objects have infinite dof! Our Approach: Build roadmap relaxing collision free requirement (allow penetration) Extract approximate path for a rigid version of robot (select path with smallest collision/penetration) Follow path and deform the robot to avoid the collisions

Deformable Objects Repairing Path using Bounding Box Deformation build a 3D voxel bounding box. Convert it to ChainMail bounding box (3D grid of springs) Deform ChainMail Bounding box. Deform objects using Free Form Deformation based on deformation of the bounding box. Apply FFD ChainMail Box Deformed ChainMail Bounding Box Obstacle

Approximate Solution Query The approximate path returned is not the shortest path but the energetically most feasible path (less deformation required) Query